# How to convert multi-band Energy Time Curve (ETC) to Impulse Response (IR)?

I am running my own sound simulator. Now I have obtained the Energy Time Curve (ETC, expressed by energy) by physical simulation, and they are stored in 8 discrete frequency bands (because decay of sound in the air is dependent on sound frequency). Now I am having problem getting the impulse response (IR, expressed by amplitude) of the measured system. I will need the IR to convolve with some audio files.

So I have $ETC(f,t)$, where $f\in\{f_1,f_2,...,f_8\},t=1,2,...N$. And the length $N$ is sufficient to capture the energy decay under a sample rate of 48kHz. How should I derive $IR(t)$ from the above $ETC$?

Thanks in advance.

• i think you need to be more explicit about what you mean by the ETC. i remember the Heyser days and the ETC was the square of the envelope of what EEs call the analytic signal. is it that. something like $\mathrm{ETC}(f,t)$ looks to me to be something like the magnitude square of the STFT. with the latter, you lose phase information and an All-Pass Filter and a delay line and even a wire will look the same (but they have different IR). – robert bristow-johnson Jul 21 '17 at 5:56
• I know of people who have good results using RAM PE in air acoustics staff.washington.edu/dushaw/AcousticsCode/…. I recall that there is an example of propagating a DFT of a broadband signal and reconstructing it. I'm not familiar with the term ETC so I'm guessing that is similar to your problem – user28715 Jul 23 '17 at 4:53
• Does your ETC have a form of echograms? If it does then you can synthesize the IR from band-pass echograms. This paper contains a very simple method to do that. – jojek Jul 4 '20 at 15:15

## 1 Answer

You can't really get the impulse response from the ETC. This is because ETC represents the energy of the signal, while the impulse response contains time information too.

One may argue that ETC also has some time information but it's not the same. You need phase to be able to reconstruct the impulse response and this is simply not present in the ETC.

What you could possibly try to do is to sum the energy of the frequency bands to acquire the total energy (broadband), calculate the square root of the result to acquire the amplitude and add random phase, or even impose the phase characteristics you want. Keep in mind that this may lead to unstable results or complex values in your time series (IR).

In order to impose your desired phase response, you could convolve your signal (square root of total energy) with the IFFT of a "manufactured" signal that will have constant amplitude spectrum and the desired phase.

Once more, please keep in mind that arbitrary phase curves may lead to results that "do not make sense". No matter what, the resulting sound may, or may not sound natural/real. Most probably (haven't tested though) parts of it, such as the late reverberation tale may sound natural (it exhibits random-like phase characteristics) while the fist parts of the IR may not sound convincing.

The main issue with the manufactured phase is that real spaces exhibit mixed-phase characteristics. Some parts of the IR (early reflections) have minimum phase (most often) while late reverberation shows maximum phase characteristics.